Locally maximal embeddings of graphs into surfaces (Seminar DM)

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Locally maximal embeddings of graphs into surfaces

Michal Kotrbčík, Comenius University, Bratislava, Slovakia

Torek, 26. oktober 2010, od 10-12, Plemljev seminar, Jadranska 19

A locally maximal embedding of a graph G is a cellular embedding of G
on an orientable surface such that every vertex of G is incident with at
most two faces. These are precisely the embeddings whose genus cannot be
raised by moving any edge in the local rotation at one of is end-vertices.
The locally maximal genus of a graph G is the minimum genus of a locally
maximal embedding of G. We investigate locally maximal embeddings of
graphs, providing a method for constructing locally maximal embeddings
with many faces (that is, with low genus). We show that the maximum number
of vertex disjoint cycles of G is a tight upper bound on the maximum
number of faces of any locally maximal embedding of G. By combining
these two results we are able to calculate the locally maximal genus of
many interesting classes of graphs, for example complete graphs, complete
bipartite graphs and hypercubes. We investigate the relationships between
locally maximal genus, minimum genus, and maximum genus. In particular, we
provide results towards a classification of all graphs with locally
maximal
genus equal to minimum genus.